The version of the Code in effect for this home will also have an important impact on the replacement of the grounding type receptacle devices.
Existing multiwire branch circuits will require two pole GFCIs, and any cross-neutrals or existing neutral - EGC connections are going to require finding and correcting.
If either the 2014 or 2017 are in effect, the act of replacement invokes all of 406.4(D). The hard one "may" be 406.4(D)(4) and the requirement of AFCI. Even though you are going back to nongrounding type receptacle devices, it IS replacing the device. Now, the inspector is wrong to suggest going back to nongrounding type receptacles, as the GFCI protected grounding type receptacle without an EGC is compliant.
Also, there is unfinished business for receptacles that have certain equipment installed where the equipment manufacturer's instructions require an Equipment Grounding Conductor (EGC) connection for their cord (think microwave, refrigerator, window AC, cable or satellite set top box, etc.) so the GFCI protection doesn't help with that.